# Number of principle solutions of the equation

• MHB
• DaalChawal
In summary, there are two principal solutions for the equation $sinx + cosx + sin2x + cos2x + sin3x = -1$ when solved without using Desmos.
DaalChawal
Find number of principal solutions of the equation
$sinx + cosx + sin2x + cos2x + sin3x = -1$
Please help anyone wihout using desmos

There are a few different ways to approach this problem without using Desmos. One method is to rewrite the equation using trigonometric identities to simplify it. For example, we can rewrite the equation as:

$sinx + cosx + 2sinxcosx + 2cos^2x - 1 = -1$

Then, we can use the double angle formula for cosine to rewrite the equation as:

$sinx + cosx + 2sinxcosx + 1 - 2sin^2x = -1$

Next, we can rearrange the terms to get:

$2sinxcosx - 2sin^2x = -1 - sinx - cosx$

Now, we can use the Pythagorean identity for sine to rewrite the equation as:

$2sinxcosx - 2(1-cos^2x) = -1 - sinx - cosx$

After simplifying, we get:

$2cos^2x + 2sinx - 1 = 0$

This is now a quadratic equation in terms of cosx. We can use the quadratic formula to solve for cosx:

$cosx = \frac{-2 \pm \sqrt{4 - 4(2)(-1)}}{2(2)}$

Simplifying further, we get:

$cosx = \frac{-2 \pm \sqrt{12}}{4}$

$cosx = \frac{-2 \pm 2\sqrt{3}}{4}$

$cosx = \frac{-1 \pm \sqrt{3}}{2}$

Now, we can plug these values back into the original equation to solve for x. We get two possible solutions:

$x = \frac{\pi}{3} + 2\pi n$ or $x = \frac{5\pi}{6} + 2\pi n$ where n is any integer.

Therefore, there are two principal solutions for the given equation.

## 1. What is the definition of "number of principle solutions of the equation"?

The number of principle solutions of an equation refers to the total number of distinct values that satisfy the equation. It is also known as the number of roots or solutions of the equation.

## 2. How do you find the number of principle solutions of an equation?

The number of principle solutions can be found by solving the equation and counting the distinct values that satisfy it. The degree of the equation (highest power of the variable) also gives an upper limit on the number of principle solutions.

## 3. Can an equation have more than one principle solution?

Yes, an equation can have more than one principle solution. This is especially true for equations with a higher degree, such as quadratic or cubic equations.

## 4. What is the significance of the number of principle solutions in an equation?

The number of principle solutions in an equation gives important information about the behavior of the equation and the graph of the equation. It can also help determine the number of intersections or points of intersection between two equations.

## 5. Are there any special cases where an equation may have an infinite number of principle solutions?

Yes, there are special cases where an equation may have an infinite number of principle solutions. For example, a linear equation with a slope of 0 will have infinitely many solutions, as all points on the x-axis will satisfy the equation.

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